Why is snap bone to cursor not working in pose mode?
[Clamp To] Constraint  wins over manual placement such as [Snap To].


An active [clamp to] constraint of influence 1, with a disconnected bone, will dominate any manual position for final placment.  This is useful, designed, and meets the expectation of the Blender user who is informed.  Snap to is manual placement.  Standard specification and behavior.
Verify your bone is NOT connected to parent in bone edit mode.  Then the bone should follow the curve.





You may change your curve to be pass through any location including the cursor location.  Suit to your needs.
Be aware the constraints can be turned off and on with the eye option, and their influence can be set from 0 to 1 on the constraint panel.  Both of these options can use animation keyframes. So if you reduce the influence to zero with keyframes the bone will be off the curve for those keyframes.  Thus you can have a bone follow the curve or be off the curve as suits your animation.





If a constraint is used to control placement it dominates over manual placement.  That is the designed behavior.